Now, to generalize the above discussion to the case where the PCF \(\Gamma _i\)  is not necessarily zero for some \(i=1,\ldots ,N\) , we consider the following multiparameter eigenvalue problem and its generalization (instead of the standard eigenvalue problem (REF )).
The multiparameter eigenvalue problem (MEP) [1]}, [2]} is to find a value \(\lambda \in \mathbb {C}\) , a sequence \(\lbrace a_{i,k}\rbrace _{i,k=1}^N\)  in \(\mathbb {C}\) , and a vector \(\mathbf {u}=[u_1,\ldots ,u_N]\in \mathcal {H}\)  such that
\(\lambda ^{-1}K\mathbf {u}=\sum _{i,k=1}^Na_{i,k}B_{i,k}\mathbf {u}.\) 
